144 XIII INTEGRATION (b) Suppose that, for each s > 0, there exists a measurable set A, an integrable func- tion g ^ 0, and an integer n0 such that, for all n^>n0, we have I \fn\ dp ^ e and !/«(*)! ^ \d(*)\ for all xeA. Show that /is integrable and that lim f I/-/J dp,= 0. n-*oo J Consider the converse. (c) Show by examples that the conditions of (a) are not sufficient and that the con- ditions of (b) are not necessary for /to be integrable and \fdp. = lim \fn dfi. J n-*oo J 5. If A is measurable and B is any subset of X, show that ^*(B) = ^*(B n A) + /z*(B n (X -A)). (If /u*(B) < + oo, consider an integrable set Bx => B such that /x*(B) = ft(Bi).) Con- versely, show that if A satisfies this condition (for all B <= X), then A is measurable (cf. Section 13.8, Problem 3). 6. Suppose that X is compact. A bounded real-valued function / on X is said to be continuous almost everywhere (with respect to /x) on X if the set of points of discon- tinuity of /is negligible. (a) Give an example of a function /which is continuous almost everywhere and such that there exists no continuous function g which is equal to /almost everywhere. (b) Suppose that the support of ft is equal to X. Show that a bounded real-valued function /defined on X is equal almost everywhere to an almost everywhere contin- uous function on X if and only if there exists a subset A of X such that X — A is negligible and f\ A is continuous. (To show that the condition is sufficient, observe that A is dense in X and that the lower semicontinuous extension of/| A to X is continuous at every point of A.) Deduce that /is measurable. Show that there exists a sequence (/,) of continuous functions on X which converges at every point of X and whose limit is almost everywhere equal to/(cf. Section 13.11 , Problem 3). (c) Show that a real-valued function / on R which is continuous on the right (i.e., such that/fc-f ) —f(x) for all x e R) is continuous except at the points of an at most denumerable. set, and is therefore continuous almost everywhere with respect to Lebesgue measure. (Apply Section 3.9, Problem 3 to the set An of points x e R at which the oscillation of /is >!/«.) 7. A partition w = (A«) of a set E is said to be finer than a partition w' = (A'ff) of E if, for each index a, there exists ft such that A« <= A^ . The partitions of E form an ordered set with respect to this relation. (a) Let X be a compact metric space. For each finite partition w = (Afc) of X con- sisting of integrable sets and each bounded real-valued function /on X, put Sw(/) - sup k xeAk O'Riemann sums" relative to /and the partition TZJ). Show that and that if w is finer than w'9 then sw>(f) <; sw(f) and Sro(/) g Sw>(/).]£ hm <*g